Download Form groups of two or three and discuss the following questions

Module H1 Practical 8
Probabilities associated with the Normal Distribution
1. Suppose a reference distribution for age-specific weight, for a particular age group of
children, is normal with mean 10kg and variance 2.25kg2. Suppose also that children with
z-scores < - 2 are regarded as being moderately to severely underweight, while those with z-scores
< -3 are regarded as being severely underweight. Classify the following weights of children as
to whether they are not underweight, moderately underweight and severely underweight
8.8
10.3
7.5
9.2
11.4
6.9
7.7
9.5
8.3
7.9
9.9
6.7
7.8
7.7
8.6
10.5
How many fall into each of the three categories above? Note down your answers below.
Number not underweight:
Number moderately underweight:
Number severely underweight:
2. Suppose fuel consumption per 100 Km of a 4x4 truck is normally distributed with
mean 18.3 litres and standard deviation 1.2 litres.
Defining a suitable random variable X, calculate the probability that on a particular 100 Km
stretch the truck would need more than 20 litres.
(a) First use tables of the standard normal distribution for this purpose, and note down any
computations you do, and your final answer below.
SADC Course in Statistics
Module H1 Practical 8 – Page 1
Module H1 Practical 8
(b) Verify your answer using the Excelfunction
NORMDIST(x, mean, standard deviation, cumulative)
where cumulative=TRUE gives the cumulative distribution, i.e. area under the normal
curve from - to x, while cumulative=FALSE gives the value of the normal probability
density function. (Note: You will rarely need the latter).
Thus, for example, NORMDIST(x, mean, standard deviation, cumulative) = Pr(X<x).
Note also: NORMSDIST(z) = NORMDIST(x, 0, 1, TRUE).
Using either of the above functions, what is Pr(X>20)?
(c) Suppose you do not want to take a risk of more than 1 in a 100 of running out of fuel.
What is the minimum amount of fuel that you should put in your tank for a 100 Km trip?
Find an answer to this question using statistical tables, then explore the use of the function
NORMINV to verify your answer. Note down the steps you undertook below, and write
down your final answer.
SADC Course in Statistics
Module H1 Practical 8 – Page 2
Module H1 Practical 8
3. IF YOU HAVE TIME, produce a histogram similar to that shown on slide 4 of the
lecture session. This would be useful to visually judge whether a given variable follows a
normal distribution according to how close the pattern of the histogram would be to a bellshape. Note however that this approach is applicable only if you have a sufficiently large
sample size, e.g. more than about 50 observations. Another alternative, suitable for small
sample sizes, is presented in the next session.
Instructions to produce a histogram in Excel are as follows:
Open the Excel workbook named H1_data.xls and move to the sheet named Normal.
This contains data for the weights of 100 maize cobs.
(a) First create a new column called (say) intervals, with values starting at 80 and
incrementing in steps of 7 until reaching the value 199, i.e. 80, 87, 94, etc. These values are
suggested so that the intervals are as close as possible to the graph you saw on slide 4.
(b) Now use the menu sequence Tools, Data Analysis, and select Histogram. In the
resulting dialogue, give the Input Range as A2:A101 and the Bin Range as B2:B19. Click OK.
(c) Make the resulting bar chart you see a bit bigger, e.g. longer from top to bottom.
Notice that Excel produces bar chart, but a histogram should not have gaps between the
bars since the variable being represented (weights) is a continuous variable.
(d) To remove the gaps, click on any bar. This should highlight just the bars only. Then
right click and chose Format Data Series.
(e) Go to Options in the resulting dialogue and make the gap width=0. Click OK. This
should give you a histogram closely resembling that shown on slide 4 of the lecture session.
Do you think the graph has the shape of a normal distribution? You could check this
comparison by using SSC-stat1 on the menu, then Visualisation, Density Estimate, while being
on the raw data sheet named Normal, and selecting cobwts as the variable. The resultant
graph shows the outline of the data histogram, along with a smoothed normal curve.
1
If you do not see SSCstat on your tool bar, try Tools, Add-Ins… and click against SSC-Stat. If this
option is not shown as an add-in, then follow instructions given in the Intermediate Level Module I2 to
download this add-in.
SADC Course in Statistics
Module H1 Practical 8 – Page 3